Determining triple half-twist of optical Möbius strips in electromagnetic fields with a non-planar structure

Polarization ellipses in a nonparaxial field form twisted strips when traced along closed contours around a circular polarization singularity line. We found an analytical expression for the twist number of the strip when the contour is coplanar with the polarization ellipse in its center. Necessary and sufficient conditions for strips having one or three half-twists are found. A set of five parameters of electromagnetic field at the polarization singularity point is found which definitely determines the value of the twist number of the strip.